Formation of Images by Spherical Mirrors Spherical mirrors are shaped like sections of a sphere, and may be reflective on either the inside (concave) or outside (convex). Figure 32-10. Mirrors with convex and concave spherical surfaces. Note that θr = θi for each ray.

Formation of Images by Spherical Mirrors Rays coming from a faraway object are effectively parallel. Figure 32-12. If the object’s distance is large compared to the size of the mirror (or lens), the rays are nearly parallel. They are parallel for an object at infinity (∞).

Formation of Images by Spherical Mirrors Parallel rays striking a spherical mirror do not all converge at exactly the same place if the curvature of the mirror is large; this is called spherical aberration. Figure 32-13. Parallel rays striking a concave spherical mirror do not intersect (or focus) at precisely a single point. (This “defect” is referred to as “spherical aberration.”)

Formation of Images by Spherical Mirrors If the curvature is small, the focus is much more precise; the focal point is where the rays converge. Figure 32-14. Rays parallel to the principal axis of a concave spherical mirror come to a focus at F, the focal point, as long as the mirror is small in width as compared to its radius of curvature, r, so that the rays are “paraxial”—that is, make only small angles with the horizontal axis.

Formation of Images by Spherical Mirrors Using geometry, we find that the focal length is half the radius of curvature: Spherical aberration can be avoided by using a parabolic reflector; these are more difficult and expensive to make, and so are used only when necessary, such as in research telescopes.

Formation of Images by Spherical Mirrors We use ray diagrams to determine where an image will be. For mirrors, we use three key rays, all of which begin on the object: A ray parallel to the axis; after reflection it passes through the focal point. A ray through the focal point; after reflection it is parallel to the axis. A ray perpendicular to the mirror; it reflects back on itself.

32-3 Formation of Images by Spherical Mirrors Figure 32-15. Rays leave point O’ on the object (an arrow). Shown are the three most useful rays for determining where the image I’ is formed. [Note that our mirror is not small compared to f, so our diagram will not give the precise position of the image.]

Formation of Images by Spherical Mirrors The intersection of these three rays gives the position of the image of that point on the object. To get a full image, we can do the same with other points (two points suffice for may purposes).

Formation of Images by Spherical Mirrors Geometrically, we can derive an equation that relates the object distance, image distance, and focal length of the mirror: Figure 32-16. Diagram for deriving the mirror equation. For the derivation, we assume the mirror size is small compared to its radius of curvature.

Formation of Images by Spherical Mirrors We can also find the magnification (ratio of image height to object height): The negative sign indicates that the image is inverted. This object is between the center of curvature and the focal point, and its image is larger, inverted, and real.

Formation of Images by Spherical Mirrors Example : Image in a concave mirror. A 1.50-cm-high diamond ring is placed 20.0 cm from a concave mirror with radius of curvature 30.0 cm. Determine (a) the position of the image, and (b) its size. Solution: a. Using the mirror equation, we find di = 60.0 cm. b. Using the magnification equation, we find M = -3.00 and hi = -4.5 cm.

Formation of Images by Spherical Mirrors Conceptual Example : Reversible rays. If the object in this figure is placed where the image is, where will the new image be? Figure 32-16 goes here. Solution: The equations, and the physical setup, are symmetric between the image and the object. The new image will be where the old object was.

Formation of Images by Spherical Mirrors If an object is outside the center of curvature of a concave mirror, its image will be inverted, smaller, and real. Figure 32-18. You can see a clear inverted image of your face when you are beyond C (do > 2f), because the rays that arrive at your eye are diverging. Standard rays 2 and 3 are shown leaving point O on your nose. Ray 2 (and other nearby rays) enters your eye. Notice that rays are diverging as they move to the left of image point I.

Formation of Images by Spherical Mirrors Example : Object closer to concave mirror. A 1.00-cm-high object is placed 10.0 cm from a concave mirror whose radius of curvature is 30.0 cm. (a) Draw a ray diagram to locate (approximately) the position of the image. (b) Determine the position of the image and the magnification analytically. Figure 32-17. Object placed within the focal point F. The image is behind the mirror and is virtual, [Note that the vertical scale (height of object = 1.0 cm) is different from the horizontal (OA = 10.0 cm) for ease of drawing, and reduces the precision of the drawing.] Example 32–6. Solution: a. The figure shows the ray diagram and the image; the image is upright, larger in size than the object, and virtual. b. Using the mirror equation gives di = -30.0 cm. Using the magnification equation gives M = +3.00.

Formation of Images by Spherical Mirrors For a convex mirror, the image is always virtual, upright, and smaller. Figure 32-19. Convex mirror: (a) the focal point is at F, behind the mirror; (b) the image I of the object at O is virtual, upright, and smaller than the object.

Formation of Images by Spherical Mirrors Problem Solving: Spherical Mirrors Draw a ray diagram; the image is where the rays intersect. Apply the mirror and magnification equations. Sign conventions: if the object, image, or focal point is on the reflective side of the mirror, its distance is positive, and negative otherwise. Magnification is positive if image is upright, negative otherwise. Check that your solution agrees with the ray diagram.

Formation of Images by Spherical Mirrors Example : Convex rearview mirror. An external rearview car mirror is convex with a radius of curvature of 16.0 m. Determine the location of the image and its magnification for an object 10.0 m from the mirror. Solution: The ray diagram for a convex lens appears in Figure 32-19b. A convex mirror has a negative focal length, giving di = -4.4 m and M = +0.44. The image is virtual, upright, and smaller than the object.